Discounted Stock Prices and Martingales Under the Physical Measure

Summary

The document poses a question about whether a discounted stock price that is a martingale under a risk-neutral measure must fail to be a martingale under the actual or physical measure. It frames the issue in a Black–Scholes setting, with a geometric Brownian stock price and a bank account accruing at a possibly time-varying short rate.

The text provides no answer, derivation, or empirical evidence, so it does not establish how the discounted price behaves under the physical measure. Its useful contribution is limited to identifying a foundational distinction in asset pricing: martingale properties depend on the chosen probability measure and on the drift assumptions. Further analysis would need the stock’s physical-measure dynamics and the relationship between the physical and risk-neutral measures.

Key ideas

• A discounted stock price is specified to be a martingale under a risk-neutral measure.
• The document asks whether that property also holds under the physical probability measure.
• It situates the question in a Black–Scholes model with a geometric Brownian stock and an accruing bond.
• No answer or derivation is included, so the physical-measure behavior remains unresolved.

Tags

# Discounted stock price under a NON risk-neutral measure


# Discounted stock price under a NON risk-neutral measure












Under a risk-neutral measure $\mathbb{Q}$, the discounted stock price is a $\mathbb{Q}$-martingale. Does it mean that under the actual probability measure $\mathbb{P}$ the discounted stock price is NOT a $\mathbb{P}$-martingale? Assuming that we are working with a Black-Scholes market model where the stock price $S(t)$ is a geometric Brownian motion and the bond dynamic is given by

$dB(t) = r(t)B(t)dt$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.